TEACHING PHILOSOPHY
Fear of mathematics means that professors of mathematics have a great responsibility in imparting math knowledge to the next generation. The vast majority of students who begin a mathematics class wonder if this math class will be the one blockade preventing them from reaching their future goals. Students know the work in a math class is going to be difficult; furthermore, they do not enjoy the luxury of knowing that as long as they try their hardest, then everything will turn out alright. As Euclid once said, “There is no royal road to geometry.” Fearing the fearful is warranted and recognizing justified fear is the first step in facing an arduous task. Even though math anxiety is a genuine concern, it is by no means a permanent affliction.
My teaching philosophy revolves around dealing with the fear of mathematics. Students that are viewed as lazy are more often dealing with a large list of priorities, all of which drain their time and energy. The number one cause of inability to succeed in a math class comes from the fear generated by not knowing where to start an assignment, not knowing what to ask, and the fear that the great investment of time required to learn mathematics will not pay off in the end. Time is a valuable commodity in college and time lost cannot be regained. Large investments of time prevent the depressed student from ever opening his/her math book as it is more comforting to leave the math book unopened or in some cases even un-purchased.
In no way should a professor lie to a student by insinuating that math is easy and exciting at every moment. If math were easy, then everyone would do it. Students need to hear about the effort it takes to succeed, that perseverance is not always fun, and that instant gratification is a poison. Students need to know that hard work pays off and that the rewards are well worth the time and effort put into analytical investigations.
My work involves reducing the genuine and the perceived fear of the student. I have to present material in a manner that produces student confidence and that demonstrates a genuine study process will indeed produce attainment. My presentations produce self-assurance without compromising the fidelity of the curriculum. Student understanding is the catalyst to eliminating student fear. Learners begin to excel in mathematics when they find the joy of understanding and the aesthetic beauty inherent in mathematics.
My job is to impart the importance of effort in learning how to learn. Solutions are merely the end goal of a learned thought process. The importance of education is in the stretching of the intellect and the exercising of the mind, i.e., the development of a learning method. The type of thinking necessary for a mathematician is not un-like that of a poet: creative and unique. The reasoning demonstrated by mathematicians produces ideas, new theories, and solutions to previously unanswered questions. This type of thinking cannot always be demonstrated on a 50-minute exam. Often the most creative thinking occurs in an off moment while taking a stroll, driving the car, or even taking a shower. As such, I do not expect students to perform innovative thinking on the spot during an exam while a clock ticks above their head. Instead I ask them to demonstrate an understanding of the question and to demonstrate effort in their explanation. I ask them to be able to communicate their understanding in a neat and organized manner. As Einstein once commented, “Any fool can know, the point is to understand” and “You do not truly understand something unless you can explain it to your grandmother.” By imparting my goal of understanding to the student, I provide students with confidence and a clear roadmap for their studies.
Individual math problems, as found in nearly all textbooks, are limited in their effectiveness for providing problem solving abilities. I find it more important to study solution processes and this often means working from solution to question rather than from question to solution. This process is not unlike studying the work of the great artists before attempting one’s own creation. Remedial students, especially, find a version of this idea particularly helpful. In education circles the process is known as “dependent learning with a focus on procedure.” Demonstrating solution methods to students and allowing them to work backwards as well as forwards provides them with material on which to build their own creative works.
Although numerous mathematics professors assign copious amounts of homework to be collected and graded, I do not. I make it clear that homework is merely a form of studying and that not doing homework is paramount to not studying. Mathematics can only be learned with practice. Math is not innately known; it is a learned ability, honed with drill and repetition. The homework that I hope to collect from students is work that is perhaps unfinished or even done incorrectly. This allows them to understand what they need help with and it allows me to see where they need help. At this juncture I am able to help them grow, and they realize that they are gaining new insights into mathematics. I am happy to provide answers to homework since student efforts are towards understanding. Handing in a perfect paper shows me that I am not “teaching” anything; rather I am putting the student through a tiresome exercise. By way of analogy, if I were a doctor and the patient (student) hurt his/her finger, I would not want to see the healthy hand, I want to see the injured hand.
Since classrooms typically contain students at various skill levels, I try to present questions of varying difficulty. Proper assessment of the students is a delicate balancing act. I must be able to ask questions that challenge the diverse abilities in class but not ask questions that will crush the spirit of those who are just starting to comprehend. In my approach to mathematics, I attempt to do away with much of the fear associated with being asked to answer a seemingly impossible question on a high-stakes test while a clock is ticking in the background. As long as the student has done his/her work the majority of this fear is abated.
The usual enigma for the math student to decipher is determining what knowledge is of most importance. Where should the student put his/her time and effort? With time at a premium, students will study only what is required and no more. My job is to provide a clear picture of what is necessary. Non-essential material is not presented in a haphazard manner with the expectation that it will be on an assessment. Concepts are presented and expected to be mastered or they are not covered at all. Students too often perform poorly on an examination because they fear the study process. Students must know what is expected of them.
Student learning progresses with note taking. I believe it is of vital importance in mathematics for students to create their own notes during lecture. The physical action of writing, hand-eye coordination, and time consuming drawing instills the student with a bit of the knowledge requisite in performing mathematics. I do not make power-point notes before class. Transparencies and/or projected information have their place, but in a math class ready-made notes are best used only to relate data or display a figure. Spontaneous lecture, in a face-to-face setting, cannot evolve appropriately with pre-made notes. Though I do prepare assiduously for my lectures, I do not read from notes as this sets before the student the concept that math cannot be learned, it is merely copied from others.
Students know very well that answering math questions is difficult and that asking questions is even more difficult. Asking for help often promotes a feeling of helplessness and even foolishness. I make it a point to be available at any time in order to put the student at ease. Office hours are the minimum requirement. My students can contact me at any time via phone, text, or email. This often turns out to be the reality since students have schedules that are anathema to office hours during the school day. The modern Southwestern student is often going to another class, commuting, or working until late at night. I believe communication at all times is in the best interest of the student.
Communicating mathematics is a precise art. Lazy or sloppy use of language makes mathematics much harder to learn. Once students have developed an incorrect understanding, they are 10 times harder to teach and their learning is hampered to an unknown degree. Un-teaching a faulty idea is much harder than teaching a correct concept. When meaningless phrases are used to communicate ideas (“cancel out,” “goes away,” “put together,” etc.) then students often develop their own reasons and meaning for the phrases. Invalid and incorrect language increases the work of the student. I insist on using correct language in all descriptions given either by my students or given by myself. Presentation of material must be unambiguous and trustworthy.
I instill in my students the awareness that they do not fully understand an idea until they can write about it in a coherent manner. Writing a proof requires numerous revisions – from the skeleton of the idea, to the writing of the proof (a dozen times over), to the final draft written with proper grammar and notation written in a succinct and intelligible style. The greatest way to learn math is to be able to teach it to others. No other way comes close. Communicating an idea to another person will make the idea comprehensible and permanent for the presenter.
One of my undergraduate degrees is in engineering and the other has a computer emphasis. I enjoy technology and I fully understand its limitations and its downfalls. The misuse of technology can, and has, permanently damaged a number of students’ learning abilities. Even though the tools of technology are indispensable, they are not needed at every moment and in every instance. Learning is handicapped when an academically undeveloped student forgoes understanding for the quick fix found in technology. The role of new computer algebra systems is vital for furthering knowledge, but they are simultaneously harmful to the purpose of mathematical education. Instruction in the proper use of computer algebra systems is another key to quality mathematical education.
Mathematics is a tool for the other disciplines and mathematics is present in daily society. As a result, application of math topics is a common direction followed by a number of mathematics instructors. I focus on a minimum of application so as not to water down the understanding of mathematical principles. Application is important but not at the expense of course objectives. The majority of application is found in a course that involves the actual application, in this venue the application is done proper justice. In other words, mathematical principles are best put to use when application presents itself.
In the end, I feel that I have no better daily advice for myself than to know my student. I must fully understand who my students are, where they come from, and where they expect to be in the future. Though I do not attempt to be their best friend, or their father figure, I am their teacher whose end game is to make sure they learn mathematics. I can only do this by being fully aware of every student’s strengths, weaknesses, and interests. When I understand their motives, and when I can provide them with purpose, then I can effectively monitor their enthusiasm to continue their endeavors. Ultimately, nothing I say or do will make a difference if the student is unmotivated to implement my advice and/or instruction.
Mathematics is a challenging discipline. Students of mathematics find it difficult, professional mathematicians find it difficult, and professors of mathematics find it difficult. I too find mathematics to be a demanding discipline. Learning mathematics is taxing, complicated, and intimidating to every person who is candid about hi/her feelings. Even the most accomplished mathematician regards mathematics with respect and a bit of fear.